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On the spectrum of the Cesaro operator. (English) Zbl 0548.47017
The Cesàro operator $$Cx=y$$ where $$y_ n=\frac{x_ 1+x_ 2+...+x_ n}{n}$$ is shown to have spectrum $$|\lambda -{1\over2}|\leq {1\over2}$$ when acting on the space $$c_ 0$$ of sequences convergent to zero. C is shown to have no eigenvalues, whilst its adjoint $$C^*$$ has eigenvalues $$|\lambda -{1\over2}| <{1\over2}$$ all simple. The methods are similar to those of Halmos et. al. dealing with C on $$\ell^ 2$$ (reference given) except that a direct proof of the invertibility of C-$$\lambda$$ I for $$|\lambda -{1\over2}| >{1\over2}$$ is needed.

##### MSC:
 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A10 Spectrum, resolvent 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
##### Keywords:
Cesàro operator; spectrum; eigenvalues
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